What is the name of this inequality involving sums of squares and rectangles?

  • Context: Undergrad 
  • Thread starter Thread starter Noesis
  • Start date Start date
  • Tags Tags
    Inequality
Click For Summary

Discussion Overview

The discussion revolves around an inequality involving sums of squares and products of numbers, with participants exploring its properties, potential names, and related concepts. The scope includes mathematical reasoning and exploration of inequalities.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that the inequality, expressed as a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1, may have a geometric interpretation where the sum of squares is greater than the corresponding sum of rectangles.
  • Others challenge the validity of the inequality, providing counterexamples such as setting a = b > c, which leads to a situation where the left-hand side equals 2ab while the right-hand side does not hold true.
  • One participant mentions that if all numbers are equal, the left-hand side equals n, while the right-hand side equals n(n-1), indicating a discrepancy.
  • A participant references the rearrangement inequality, suggesting it is a related concept that applies when rearranging two sets of numbers.
  • Another participant offers a tutorial perspective, indicating that to correctly apply the inequality, one must omit half of the terms and relates it to the general theorem for vector spaces with inner products.
  • There is mention of the Cauchy-Riemann-Schwartz inequality and its connection to the discussion, suggesting that it may yield similar results or insights.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the proposed inequality, with some supporting its geometric interpretation while others provide counterexamples that challenge its correctness. No consensus is reached regarding its established name or validity.

Contextual Notes

Limitations include the need for clarity on assumptions regarding the values of a, b, and c, as well as the specific conditions under which the inequality may hold or fail. The discussion also highlights the dependence on definitions and interpretations of the terms involved.

Noesis
Messages
99
Reaction score
0
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.
 
Mathematics news on Phys.org
Noesis said:
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.

It doesn't look like it's true. Let a=b>c. Then a2+b2=2ab, while c2<2(ac+bc).
 
Noesis said:
Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.
There are a lot more of the rectangles than there are the squares... in fact, another easy source of counterexamples is just to make all the numbers the same! e.g. if they're all 1, then the L.H.S. is n, but the R.H.S. is n(n-1)
 
A very similar inequality is called the rearrangement inequality which works when you rearrange two sets of number, not one.
 
Noesis said:
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.

Short tutorial on the subject.

First to get it right you need to omit half the terms.
a2 + b2 + c2 >= ab + bc + ac
There is a general theorem for vector spaces with inner (dot) products, namely
A.B=|A||B|cosx, where x is the angle between A and B.

For the case you are interested in:
A=(a1,a2,a3,...,an) while B=(a2,a3,...,an,a1)
 
mathman said:
Short tutorial on the subject.

First to get it right you need to omit half the terms.
a2 + b2 + c2 >= ab + bc + ac
There is a general theorem for vector spaces with inner (dot) products, namely
A.B=|A||B|cosx, where x is the angle between A and B.

For the case you are interested in:
A=(a1,a2,a3,...,an) while B=(a2,a3,...,an,a1)

I had a homework assignment in my first year of undergrad asking for us to prove the Cauchy-Riemann-Schwartz inequality (a.k.a. the triangle inequality), and Googling one or both of these terms seem to generate at least a few results that have a similar form.
 

Similar threads

Proof by induction of the sum of 2 squares
  • · Replies 39 ·
2
Replies
39
Views
7K
Graduate What Are Simple Lie Algebras and How Are They Classified?
  • · Replies 1 ·
Replies
1
Views
4K
Schools How well does this course prepare me for college math?
  • · Replies 10 ·
Replies
10
Views
5K